Problems of Information Transmission, Vol. 40, No. 3, 2004, pp. 226–236. Translated from Problemy Peredachi Informatsii, No. 3, 2004, pp. 49–61.
Original Russian Text Copyright
2004 by Boiko.
METHODS OF SIGNAL PROCESSING
Wavelets and Estimation of Discontinuous Functions
L. L. Boiko
Institute for Information Transmission Problems, RAS, Moscow
Received December 8, 2003; in ﬁnal form, March 16, 2004
Abstract—The paper considers the problem of estimating a signal with ﬁnitely many points
of discontinuity from observations against white Gaussian noise. It is shown that, with an
appropriate choice of a generator polynomial, an estimation method based on wavelets yields
asymptotically minimax (up to a constant) estimates for functions suﬃciently smooth outside
the discontinuity points.
The possibility of eﬃcient numerical realization of wavelets has made them very popular in
nonparametric estimation. It is well known that in many cases statistical estimates constructed
on the basis of wavelets are optimal in the order of the convergence rate (see, e.g., [1–3]). In the
present paper, we consider one simple problem of nonparametric signal estimation, for which, with
the help of wavelets, we construct asymptotically minimax estimates up to a constant.
Assume that we observe a random process
(t)=f(t) dt +
dw(t),t∈ [0, 1], (1)
where w(t), t ≥ 0, is a standard Wiener process. Our goal is to reconstruct the signal f (t), t ∈ [0, 1],
from these observations. We consider this problem in the asymptotic setting, assuming that n →∞.
Usually, to obtain a somewhat nontrivial theory of nonparametric estimation, one deﬁnes a
class F of functions containing f(·). H¨older, Sobolev, and Besov classes are often used for this
Besides these function classes, ﬁnite-order entire analytical functions form a function class, which
is very popular in engineering applications (see, e.g., ).
This class is deﬁned as the set of real functions f such that
dξ ≤ P.
Denote it by F
(W, P ).
A remarkable property of this class, showing why it is attractive for applications, concerns the
Kotel’nikov theorem. The theorem states that, if f ∈F
(W, P ), then
sin[π(tW + k)]
π(tW + k)
2004 MAIK “Nauka/Interperiodica”